Vibroacoustic Simulation - Alexander Peiffer

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       normal upper Pi Subscript in Baseline equals upper F Subscript x Baseline ModifyingAbove u With dot (1.45)

      So we get the power balance

      that is fluctuating for harmonic motion but with a net power flow.

       StartLayout 1st Row 1st Column normal upper Pi left-parenthesis t right-parenthesis equals upper F left-parenthesis t right-parenthesis v left-parenthesis t right-parenthesis 2nd Column equals 3rd Column upper R e left-parenthesis bold-italic upper F Subscript x Baseline e Superscript j omega t Baseline right-parenthesis upper R e left-parenthesis bold-italic v Subscript x Baseline e Superscript j omega t Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column equals 3rd Column one-half upper R e left-parenthesis bold-italic upper F Subscript x Baseline e Superscript j omega t Baseline plus bold-italic upper F Subscript x Superscript asterisk Baseline e Superscript minus j omega t Baseline right-parenthesis one-half upper R e left-parenthesis bold-italic v Subscript x Baseline e Superscript j omega t Baseline plus bold-italic v Subscript x Superscript asterisk Baseline e Superscript minus j omega t Baseline right-parenthesis 3rd Row 1st Column Blank 2nd Column equals 3rd Column one-fourth upper R e left-parenthesis bold-italic upper F Subscript x Baseline bold-italic v Subscript x Superscript asterisk Baseline plus bold-italic upper F Subscript x Superscript asterisk Baseline bold-italic v Subscript x Baseline plus bold-italic upper F Subscript x Baseline bold-italic v Subscript x Baseline e Superscript j Baseline 2 omega t Baseline plus bold-italic upper F Subscript x Superscript asterisk Baseline bold-italic v Subscript x Superscript asterisk Baseline e Superscript minus j Baseline 2 omega t Baseline right-parenthesis 4th Row 1st Column Blank 2nd Column equals 3rd Column one-half upper R e left-parenthesis bold-italic upper F Subscript x Baseline bold-italic v Subscript x Superscript asterisk Baseline plus bold-italic upper F Subscript x Baseline bold-italic v Subscript x Baseline e Superscript j Baseline 2 omega t Baseline right-parenthesis EndLayout (1.47)

      The first term in the bracket is constant, the second oscillating with twice the excitation frequency. The first part is called active power and the second part the reactive. All introduced energy in one half cycle comes back in the next half cycle. The time average over one period leaves only the active part

       mathematical left-angle normal upper Pi mathematical right-angle Subscript upper T Baseline equals StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline one-half upper R e left-parenthesis bold-italic upper F Subscript x Baseline bold-italic v Subscript x Baseline Superscript asterisk Baseline plus bold-italic upper F Subscript x Baseline bold-italic v Subscript x Baseline e Superscript j Baseline 2 omega t Baseline right-parenthesis d t equals one-half upper R e left-parenthesis bold-italic upper F Subscript x Baseline bold-italic v Subscript x Baseline Superscript asterisk Baseline right-parenthesis (1.48)

      The velocity can be expressed by the impedance V=Z/F or vice versa, so we get

      The power considerations further clarify the naming conventions for the real and imaginary parts of the impedance. With Equation (1.40) the power introduced into the system equals Π=12|vx|2cv. Thus, the active power is controlled by the real part or resistance whereas the reactive part is determined by the imaginary component called reactance. The energy is dissipated in the resistive damping process, but power delivered to the reactive part goes into the kinetic and potential energy of mass and spring.

      1.2.4 Damping

      In many practical applications ζ is small and the amplitude can be estimated by linear expansion from (1.34)

       ModifyingAbove u With caret Subscript r Baseline almost-equals StartFraction ModifyingAbove u With caret Subscript 0 Baseline Over 2 zeta EndFraction left-parenthesis 1 plus StartFraction zeta squared Over 2 EndFraction right-parenthesis almost-equals StartFraction ModifyingAbove u With caret Subscript 0 Baseline Over 2 zeta EndFraction (1.50)

      with the corresponding phase angle

       phi Subscript r Baseline almost-equals arc tangent left-parenthesis minus StartFraction 1 Over zeta EndFraction right-parenthesis (1.51)

      The amplitude- and phase resonances are assumed to be equal for systems with small damping. The magnification is thus 1/2ζ, and it is called the quality factor:

       StartFraction ModifyingAbove u With caret Subscript r Baseline Over ModifyingAbove u With caret Subscript 0 Baseline EndFraction equals StartFraction 1 Over 2 zeta EndFraction equals upper Q (1.52)

      Figure 1.10 Half power bandwidth for harmonic oscillator. Source: Alexander Peiffer.

       omega Subscript 1 slash 2 Baseline equals left-parenthesis 1 plus-or-minus zeta right-parenthesis omega 0 (1.53)

      and therefore

       upper Q equals StartFraction 1 Over 2 zeta EndFraction equals StartFraction omega 0 Over omega 2 minus omega 1 EndFraction (1.54)

      Obviously the decay time is also related to damping. If Equation (1.22) is considered we get

       upper Q equals StartFraction omega 0 tau Over 2 EndFraction (1.55)

      We have presented several expressions that describe damping. Nevertheless, even more quantities for damping are used depending on the engineering discipline and will be shown in Section 1.2.5. Section 1.2.5.1 aims at sorting all those expressions and their relationships among each other.

      1.2.5 Damping in Real Systems

      Viscous damping is rare in real systems, it only exists if the surface that is connected to liquids moves so slow that no turbulent motion appears. Observation of experiments with damping normally doesn’t show damping that increases

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